Number of common points to the curves `C_(1){(-1+2cos alpha, 2 sin alpha)}` and `C_(2)(4+3sin theta,3 cos theta)` is/are equal to

Answer

To find the number of common points between the curves \( C_1 \) and \( C_2 \), we first need to express these curves in standard form.

### Step 1: Identify the curves
The curves are given in parametric form:
- \( C_1: (-1 + 2 \cos \alpha, 2 \sin \alpha) \)
- \( C_2: (4 + 3 \sin \theta, 3 \cos \theta) \)

### Step 2: Convert \( C_1 \) to Cartesian form
The parametric equations for \( C_1 \) are:
- \( x = -1 + 2 \cos \alpha \)
- \( y = 2 \sin \alpha \)

To convert this to Cartesian form, we can express \( \cos \alpha \) and \( \sin \alpha \) in terms of \( x \) and \( y \):
1. From \( x = -1 + 2 \cos \alpha \), we get:
\[ \cos \alpha = \frac{x + 1}{2} \]
2. From \( y = 2 \sin \alpha \), we get:
\[ \sin \alpha = \frac{y}{2} \]

Using the identity \( \cos^2 \alpha + \sin^2 \alpha = 1 \):
\[ \left(\frac{x + 1}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1 \]
Multiplying through by 4:
\[ (x + 1)^2 + y^2 = 4 \]
This represents a circle with center at \( (-1, 0) \) and radius \( 2 \).

### Step 3: Convert \( C_2 \) to Cartesian form
The parametric equations for \( C_2 \) are:
- \( x = 4 + 3 \sin \theta \)
- \( y = 3 \cos \theta \)

Similarly, we can express \( \sin \theta \) and \( \cos \theta \):
1. From \( x = 4 + 3 \sin \theta \), we get:
\[ \sin \theta = \frac{x - 4}{3} \]
2. From \( y = 3 \cos \theta \), we get:
\[ \cos \theta = \frac{y}{3} \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \left(\frac{x - 4}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \]
Multiplying through by 9:
\[ (x - 4)^2 + y^2 = 9 \]
This represents a circle with center at \( (4, 0) \) and radius \( 3 \).

### Step 4: Find the points of intersection
Now we have two circles:
1. Circle 1: \( (x + 1)^2 + y^2 = 4 \)
2. Circle 2: \( (x - 4)^2 + y^2 = 9 \)

To find the points of intersection, we can set the equations equal to each other:
\[ (x + 1)^2 + y^2 = 4 \]
\[ (x - 4)^2 + y^2 = 9 \]

Subtract the first equation from the second:
\[ (x - 4)^2 - (x + 1)^2 = 5 \]
Expanding both sides:
\[ (x^2 - 8x + 16) - (x^2 + 2x + 1) = 5 \]
Simplifying:
\[ -10x + 15 = 5 \]
\[ -10x = -10 \implies x = 1 \]

### Step 5: Substitute \( x \) back to find \( y \)
Substituting \( x = 1 \) into the first circle's equation:
\[ (1 + 1)^2 + y^2 = 4 \]
\[ 4 + y^2 = 4 \implies y^2 = 0 \implies y = 0 \]

### Conclusion
The common point of intersection is \( (1, 0) \). Since we have found only one common point, the number of common points to the curves \( C_1 \) and \( C_2 \) is:

\[ \text{Number of common points} = 1 \]